A.V. Bolsinov and A.T. Fomenko

Integrable Hamiltonian Systems: Geometry, Topology, Classification is essentially a research monograph and survey of recent work, covering results on the subject through the late 1990s, and quite extensive. It is terse and concise (despite its length), covering over 350 research papers, as well as recent results of the authors.

The book assumes a fairly deep familiarity with background material, and is probably meant to serve as a reference for those engaged in the field.

Michael Pearson is Director of Programs and Services of the MAA.

BASIC NOTIONSLinear Symplectic GeometrySymplectic and Poisson ManifoldsThe Darboux TheoremLiouville Integrable Hamiltonian Systems. The Liouville TheoremNon-Resonant and Resonant SystemsRotation NumberThe Momentum Mapping of an Integrable System and Its Bifurcation DiagramNon-Degenerate Critical Points of the Momentum MappingMain Types of Equivalence of Dynamical SystemsTHE TOPOLOGY OF FOLIATIONS ON TWO-DIMENSIONAL SURFACESGenerated by Morse FunctionsSimple Morse FunctionsReeb Graph of a Morse FunctionNotion of an AtomSimple AtomsSimple MoleculesComplicated AtomsClassification of AtomsSymmetry Groups of Oriented Atoms and the Universal Covering TreeNotion of a MoleculeApproximation of Complicated Molecules by Simple OnesClassification of Morse-Smale Flows on Two-Dimensional Surfaces by Means of Atoms and MoleculesROUGH LIOUVILLE EQUIVALENCE OF INTEGRABLE SYSTEMS WITH TWO DEGREES OF FREEDOMClassification of Non-degenerate Critical Submanifolds on Isoenergy 3-SurfacesThe Topological Structure of a Neighborhood of a Singular LeafTopologically Stable Hamiltonian SystemsExample of a Topologically Unstable Integrable System2-Atoms and 3-AtomsClassification of 3-Atoms3-Atoms as Bifurcations of Liouville ToriThe Molecule of an Integrable SystemComplexity of Integrable SystemsLIOUVILLE EQUIVALENCE OF INTEGRABLE SYSTEMS WITH TWO DEGREES OF FREEDOMAdmissible Coordinate Systems on the Boundary of a 3-AtomGluing Matrices and Superfluous FramesInvariants (Numerical Marks) r, e, and nThe Marked Molecule is a Complete Invariant of Liouville EquivalenceThe Influence of the OrientationRealization TheoremSimple Examples of MoleculesHamiltonian Systems with Critical Klein BottlesTopological Obstructions to Integrability of Hamiltonian Systems with Two Degrees of FreedomORBITAL CLASSIFICATION OF INTEGRABLE SYSTEMS WITH TWO DEGREES OF FREEDOMRotation Function and Rotation VectorReduction of the Three-Dimensional Orbital Classification to the Two-Dimensional Classification up to ConjugacyGeneral Concept of Constructing Orbital Invariants of Integrable Hamiltonian SystemsCLASSIFICATION OF HAMILTONIAN FLOWS ON TWO-DIMENSIONAL SURFACES UP TO TOPOLOGICAL CONJUGACYInvariants of a Hamiltonian System on a 2-AtomClassification of Hamiltonian Flows with One Degree of Freedom up to Topological ConjugacyClassification of Hamiltonian Flows on 2-Atoms with Involution up to Topological ConjugacyThe Pasting-Cutting OperationDescription of the Sets of Admissible delta-Invariants and Z-InvariantsSMOOTH CONJUGACY OF HAMILTONIAN FLOWS ON TWO-DIMENSIONAL SURFACESConstructing Smooth Invariants on 2-AtomsTheorem of Classification of Hamiltonian Flows on Atoms up to Smooth ConjugacyORBITAL CLASSIFICATION OF INTEGRABLE HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM. THE SECOND STEPSuperfluous t-Frame of a Molecule (Topological Case). The Main Lemma on t-FramesThe Group of Transformations of Transversal Sections. Pasting-Cutting OperationThe Action of GP on the Set of Superfluous t-FramesThree General Principles for Constructing InvariantsAdmissible Superfluous t-Frames and a Realization TheoremConstruction of Orbital Invariants in the Topological Case. A t-MoleculeTheorem on the Topological Orbital Classification of Integrable Systems with Two Degrees of FreedomA Particular Case: Simple Integrable SystemsSmooth Orbital ClassificationLIOUVILLE CLASSIFICATION OF INTEGRABLE SYSTEMS WITH NEIGHBORHOODS OF SINGULAR POINTSl-Type of a Four-Dimensional SingularityThe Loop Molecule of a Four-Dimensional SingularityCenter-Center CaseCenter-Saddle CaseSaddle-Saddle CaseAlmost Direct Product Representation of a Four-Dimensional SingularityProof of the Classification TheoremsFocus-Focus CaseAlmost Direct Product Representation for Multidimensional Non-degenerate Singularities of Liouville FoliationsMETHODS OF CALCULATION OF TOPOLOGICAL INVARIANTS OF INTEGRABLE HAMILTONIAN SYSTEMSGeneral Scheme for Topological Analysis of the Liouville FoliationMethods for Computing MarksThe Loop Molecule MethodList of Typical Loop MoleculesThe Structure of the Liouville Foliation for Typical Degenerate SingularitiesTypical Loop Molecules Corresponding to Degenerate One-Dimensional OrbitsComputation of r- and e-Marks by Means of Rotation FunctionsComputation of the n-Mark by Means of Rotation FunctionsRelationship Between the Marks of the Molecule and the Topology of Q3INTEGRABLE GEODESIC FLOWS ON TWO-DIMENSIONAL SURFACES 409Statement of the ProblemTopological Obstructions to Integrability of Geodesic Flows on Two-Dimensional SurfacesTwo Examples of Integrable Geodesic FlowsRiemannian Metrics Whose Geodesic Flows are Integrable by Means of Linear or Quadratic Integrals. Local TheoryLinearly and Quadratically Integrable Geodesic Flows on Closed SurfacesLIOUVILLE CLASSIFICATION OF INTEGRABLE GEODESIC FLOWS ON TWO-DIMENSIONAL SURFACESThe TorusThe Klein BottleThe SphereThe Projective PlaneORBITAL CLASSIFICATION OF INTEGRABLE GEODESIC FLOWS ON TWO-DIMENSIONAL SURFACESCase of the TorusCase of the SphereExamples of Integrable Geodesic Flows on the SphereNon-triviality of Orbital Equivalence Classes and Metrics with Closed GeodesicsTHE TOPOLOGY OF LIOUVILLE FOLIATIONS IN CLASSICAL INTEGRABLE CASES IN RIGID BODY DYNAMICSIntegrable Cases in Rigid Body DynamicsTopological Type of Isoenergy 3-SurfacesLiouville Classification of Systems in the Euler CaseLiouville Classification of Systems in the Lagrange CaseLiouville Classification of Systems in the Kovalevskaya CaseLiouville Classification of Systems in the Goryachev-Chaplygin-Sretenskii CaseLiouville Classification of Systems in the Zhukovskii CaseRough Liouville Classification of Systems in the Clebsch CaseRough Liouville Classification of Systems in the Steklov CaseRough Liouville Classification of Integrable Four-Dimensional Rigid Body SystemsThe Complete List of Molecules Appearing in Integrable Cases of Rigid Body DynamicsMAUPERTUIS PRINCIPLE AND GEODESIC EQUIVALENCEGeneral Maupertuis PrincipleMaupertuis Principle in Rigid Body DynamicsClassical Cases of Integrability in Rigid Body Dynamics and Related Integrable Geodesic Flows on the SphereConjecture on Geodesic Flows with Integrals of High DegreeDini Theorem and the Geodesic Equivalence of RiemannianMetricsGeneralized Dini-Maupertuis PrincipleOrbital Equivalence of the Neumann Problem and the Jacobi ProblemExplicit Forms of Some Remarkable Hamiltonians and Their Integrals in Separating VariablesEULER CASE IN RIGID BODY DYNAMICS AND JACOBI PROBLEM ABOUT GEODESICS ON THE ELLIPSOID. ORBITAL ISOMORPHISMIntroductionJacobi Problem and Euler CaseLiouville FoliationsRotation FunctionsThe Main TheoremSmooth InvariantsTopological Non-Conjugacy of the Jacobi Problem and the Euler CaseREFERENCESSUBJECT INDEX